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Computer program VARI-QUIR 3 provides Gauss-Seidel type of solution with inner and outer iterations for steady-state, multigroup, two-dimensional neutron diffusion equations. The program has no restrictions on any of the input parameters such as the number of groups, regions, or materials.
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This study newly proposes a heterogeneous core calculation method based on local/global iteration using proper orthogonal decomposition (POD). By using the singular value decomposition (SVD) and the low-rank approximation, appropriate POD bases for expanding the neutron flux can be obtained from snapshot data of the neutron flux obtained by fine mesh calculations. By projection using the POD bases, the dimension of the target equation (e.g., discretized neutron diffusion equation) can be dramatically reduced. In the proposed method, POD is effectively applied to each single assembly calculation (local calculation). Furthermore, using the local/global iteration, the effective neutron multiplication factor and the neutron flux distribution in the whole core geometry can be obtained by combining the numerical results of the local calculation for each fuel assembly and the global calculation for the whole core. As a feasibility study, the proposed method is applied to a one-dimensional heterogeneous core analysis, and the accuracy is investigated by changing the total number of POD bases. (authors)
This paper extends our recent work on the Physics-Informed Neural Networks (PINN) approach for the fixed source diffusion models and applies it to the diffusion theory based k-eigenvalue problems. To make the PINN equitable for the eigenvalue problems, we introduce a novel integral regularization term to the loss function in the framework, and allow the direct inference of the principal eigenvalue and the associated eigenfunction. The regularization term enforces a pre-defined value on the integration of the model predictions, and this value can be directly related to a physical property of the system. We also introduce an additional learnable parameter to approximate the principal eigenvalue. As a proof of principle, we solve the one-group two-dimensional k-eigenvalue neutron diffusion equation in this work. We then provide two numerical examples to demonstrate the applicability of the PINN approach. In each example, we solve the k-eigenvalue diffusion equation in a multi-region configuration constrained with a set of Robin boundary conditions for generality. We use a FEM solution based on the power-iteration method to verify the results of the PINN solution. The results showed relative percentage error in the predicted eigenvalue of about 0.77% and about 1.2% for example 1 and example 2, respectively. The mean absolute error in the predicted flux for example 1 is ∼ 0.002 and for example 2 is ∼ 0.0024. These results indicate some preliminary successes of the PINN application to k-eigenvalue problems. (authors)
Here, we report nuclear science and engineering is a field increasingly dominated by computational studies resulting from increasingly powerful computational tools. As a result, analytical studies, which previously pioneered nuclear engineering, are increasingly viewed as secondary or unnecessary. However, analytical solutions to reduced-fidelity models can provide important information concerning the underlying physics of a problem and aid in guiding computational studies. Similarly, there is increased interest in sensitivity analysis studies. These studies commonly use computational tools. However, providing a complementary sensitivity study of relevant analytical models can lead to a deeper analysis of a problem. This work provides the analytical sensitivity analysis of the one-dimensional (1D) cylindrical mono-energetic neutron diffusion equation using the forward sensitivity analysis procedure (FSAP) developed by Cacuci. Further, these results are applied to a reduced-fidelity model of a spent nuclear fuel cask, demonstrating how computational analysis might be improved with a complementary analytic sensitivity analysis.
In this work, we present validation test results of fully coupled neutronics and thermal-hydraulics models of the Molten Salt Reactor Experiment (MSRE) against experimental data of the zero power pump transients and the natural circulation tests at low power. To capture the strong coupling between neutronics and thermal-hydraulics due to fuel circulation, and to account for the delayed neutron precursor (DNP) distribution, the porous media thermal-hydraulics solver Pronghorn was fully coupled to the spatial neutron dynamics code Griffin, which solves the neutron diffusion equation, and to the 0-D point kinetics solver Squirrel, using a 2-D homogenized representation of the MSRE. The validation test results show very good agreement with experimental data for both point kinetics and spatial dynamics simulations, capturing the strong feedback effect and DNP losses in the MSRE. The 0-D code Squirrel accurately predicted the time-dependent behavior in the MSRE given the steady-state spatial dynamics solution of Griffin.
The extent to which suitable solutions may be obtained for one physics problem and two engineering type problems is traced. NASTRAN appears to be a practical tool to solve one-group steady-state neutron diffusion equations. Transient diffusion analysis may be performed after new levels that allow time-dependent temperature calculations are developed. NASTRAN piecewise linear anlaysis may be applied to solve those plasticity problems for which a smooth stress-strain curve can be used to describe the nonlinear material behavior. The accuracy decreases when sharp transitions in the stress-strain relations are involved. Improved NASTRAN usefulness will be obtained when nonlinear material capabilities are extended to axisymmetric elements and to include provisions for time-dependent material properties and creep analysis. Rigid formats 3 and 5 proved to be very convenient for the buckling and normal-mode analysis of a nuclear fuel element.
A general matrix method is developed for the solution of characteristic-value problems of the type arising in many physical applications. The scheme employed is essentially that of Gauss and Seidel with appropriate modifications needed to make it applicable to characteristic-value problems. An iterative procedure produces a sequence of estimates to the answer; and extrapolation techniques, based upon previous behavior of iterants, are utilized in speeding convergence. Theoretically sound limits are placed on the magnitude of the extrapolation that may be tolerated. This matrix method is applied to the problem of finding criticality and neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron fluxes in a nuclear reactor with control rods. The two-dimensional finite-difference approximation to the two-group neutron-diffusion equations is treated. Results for this example are indicated.
A diffusion-based and a discrete ordinates method are used to simulate a neutron noise experiment in the research reactor AKR-2 at the Technical University in Dresden, Germany. The AKR-2 reactor provides an interesting case for the comparison between the two methods because it is characterized by large heterogeneities and regions with low macroscopic neutron cross-sections. For the calculations, the same spatial discretization and the same set of two-energy macroscopic neutron cross-sections with isotropic scattering are used. Significant discrepancies between the diffusion-based and discrete ordinates methods are found in regions of the systems where the diffusion approximation is expected to be inaccurate in reproducing characteristics of the static neutron flux and neutron noise. (authors)
This paper presents the POLCA8 approach for modelling non-constant cross section distributions inside hexagonal fuel assemblies. The multigroup diffusion equation is modified to account for intranodal cross section variations. The obtained equation is solved in a node-wise manner based on the Fourier expansion method. As a result of varying cross sections, the solution includes a particular part additionally to the homogeneous one. A method for obtaining the particular solution is derived. Numerical tests on a VVER-1000 core are presented showing the impact of cross-section variations to some key parameters for reactor operation. (authors)
Several high-temperature gas-cooled reactor concepts (and more recently, salt-cooled designs such as the fluoride salt-cooled high-temperature reactor) feature core designs employing continuously circulating fuel pebbles. These reactor designs permit both continuous online refueling of fuel elements as well as higher overall achievable discharge burnups. However, rapid calculation of time-dependent fuel isotopic inventories proves challenging for this class of dynamic systems with current analysis tools. While iterative approaches employing coupled neutron transport have been developed to solve this issue, rapid depletion analysis techniques are needed to calculate time-dependent inventories for individual pebbles and batches (and thus the construction of full- core inventory at equilibrium). We propose a depletion analysis strategy for this type of system for cores at equilibrium. Drawing upon previous neutronic analysis of the PBMR-400 equilibrium core, we demonstrate the viability of developing collapsed one-group cross section libraries suitable for performing rapid depletion analyses with SCALE. (authors)
Dry cask storage is a method for interim storage of spent fuel assemblies which contain fissile isotopes of uranium and plutonium. These can present a proliferation concern and consequently there is a need for non-destructive testing methods to verify a dry cask's contents for proliferation protection. We present an application of adjoint sensitivity analysis and data assimilation to a multigroup diffusion model of dry cask storage. Adjoint sensitivity analysis allows the efficient calculation of sensitivities for use in data assimilation to calibrate imprecisely known parameter values and data consistency tests to detect diversion scenarios. (authors)
Ongoing design activities at Argonne National Laboratory are requiring a thorough verification of the Argonne Reactor Computation codes be performed. DIF3D is central to this system. The driver for this effort requires the 3D Cartesian, triangular-Z, and hexagonal-Z core geometry options of DIF3D be verified. Previous work identified the DIF3D features required to be verified to support current design activities, features of which are generally applicable to hexagonal-Z fast reactor designs. The scope of this verification effort includes verifying DIF3D’s ability to correctly translate the user’s model in to DIF3D’s preferred format, verifying that options planned for use have the desired effect, and verifying the correctness of the eigenvalue, fixed-source, forward, and adjoint solvers in DIF3D-FD and DIF3D-VARIANT. This manuscript provides the verification tasks and their results with respect to the features needed for current design activities. Since analytic solutions of the neutron diffusion and transport equations are either limited in scope or not possible, multiple tiers of problems unique to each solver and geometry type were implemented. Each of these tiers tests features independent and complementary arguments for why the separate testing of functionalities is acceptable. Finally, this separate testing was also supplemented with a high-level integral check of each the diffusion and transport capabilities and applicable geometries. To accommodate cases which an analytic solution is not feasible, MCNP6.2 was relied upon to provide a higher-order reference solution. This therefore required that the capabilities within MCNP6.2 which were relied upon for this work are also verified in this work. No MCNP discrepancies were noted in this effort. Note that the MCNP6.2 verification included in this work does not stand as a full verification of MCNP6.2, but merely verifies the features used in verifying DIF3D. The verification effort identified no issues that are debilitating or otherwise impactful to design usage of DIF3D, and thus DIF3D version 11.0, release 3012 is considered verified. As some additional changes have been made to the ARC software since this point all versions between release 3012 and 3266 can be considered verified as version 3253 was used for all updates in this revision. The types of issues that were identified were predominantly in the areas of: unclear documentation, software bugs which were inconsequential to final results, editing options which were ignored in favor of printing more information than requested, bugs in the outputs of intermediate results, or secondary output binary file information which was not present. While not a bug, this verification report also identified that the algorithm used to evaluate the peak fast flux in a nodal transport solution can be quite unreliable due to the methodology used and the location of the peak within the mesh. The authors of the report therefore recommend the usage of the EvaluateFlux software (distributed with ARC) as a more robust alternative noting that DIF3D will properly notify the user when the peaking values it is providing are potentially incorrect.
Ongoing design activities at Argonne National Laboratory are requiring a thorough verification of the Argonne Reactor Computation codes be performed. DIF3D is central to this system. The driver for this effort requires the 3D Cartesian, triangular-Z, and hexagonal-Z core geometry options of DIF3D be verified. Previous work identified the DIF3D features required to be verified to support current design activities, features of which are generally applicable to hexagonal-Z fast reactor designs. The scope of this verification effort includes verifying DIF3D’s ability to correctly translate the user’s model in to DIF3D’s preferred format, verifying that options planned for use have the desired effect, and verifying the correctness of the eigenvalue, fixed-source, forward, and adjoint solvers in DIF3D-FD and DIF3D-VARIANT. This manuscript provides the verification tasks and their results with respect to the features needed for current design activities. Since analytic solutions of the neutron diffusion and transport equations are either limited in scope or not possible, multiple tiers of problems unique to each solver and geometry type were implemented. Each of these tiers tests features independent and complementary arguments for why the separate testing of functionalities is acceptable. Finally, this separate testing was also supplemented with a high-level integral check of each the diffusion and transport capabilities and applicable geometries. To accommodate cases which an analytic solution is not feasible, MCNP6.2 was relied upon to provide a higher-order reference solution. This therefore required that the capabilities within MCNP6.2 which were relied upon for this work are also verified in this work. No MCNP discrepancies were noted in this effort. Note that the MCNP6.2 verification included in this work does not stand as a full verification of MCNP6.2, but merely verifies the features used in verifying DIF3D. The verification effort identified no issues that are debilitating or otherwise impactful to design usage of DIF3D, and thus DIF3D version 11.0, release 3012 is considered verified. As some additional changes have been made to the ARC software since this point all versions between release 3012 and 3266 can be considered verified as version 3253 was used for all updates in this revision. The types of issues that were identified were predominantly in the areas of: unclear documentation, software bugs which were inconsequential to final results, editing options which were ignored in favor of printing more information than requested, bugs in the outputs of intermediate results, or secondary output binary file information which was not present. While not a bug, this verification report also identified that the algorithm used to evaluate the peak fast flux in a nodal transport solution can be quite unreliable due to the methodology used and the location of the peak within the mesh. The authors of the report therefore recommend the usage of the EvaluateFlux software (distributed with ARC) as a more robust alternative noting that DIF3D will properly notify the user when the peaking values it is providing are potentially incorrect.
What is this parameter called alpha, and why is it so important? To answer these questions we consider the density of fission neutrons at any point $_r^→$ = (x, y, z) inside a volume of fissionable material. Ignoring neutron absorption, this time-varying density of neutrons is determined by two physical processes --- generation of more neutrons by fission chain reactions, and the loss of neutrons by diffusion to and through the surface. A simple, approximate mathematical description of these processes is the diffusion equation for the average neutron density n($_r^→$, t) including a fission source term.
Understanding the equation of state (EOS) of pure neutron matter is necessary for interpreting multimessenger observations of neutron stars. Reliable data analyses of these observations require well-quantified uncertainties for the EOS input, ideally propagating uncertainties from nuclear interactions directly to the EOS. This, however, requires calculations of the EOS for a prohibitively larger number of nuclear Hamiltonians, solving the nuclear many-body problem for each one. Quantum Monte Carlo methods, such as auxiliary-field diffusion Monte Carlo (AFDMC), provide precise and accurate results for the neutron matter EOS, but they are very computationally expensive, making them unsuitable for the fast evaluations necessary for uncertainty propagation. Here, we employ parametric matrix models to develop fast emulators for AFDMC calculations of neutron matter and use them to directly propagate uncertainties of coupling constants in the Hamiltonian to the EOS. As these uncertainties include estimates of the effective field theory truncation uncertainty, this approach provides robust uncertainty estimates for use in astrophysical data analyses. In conclusion, this Letter will enable novel applications such as using astrophysical observations to put constraints on coupling constants for nuclear interactions.
SIMULATE5-K is Studsvik's next generation best estimate transient code. The time dependent diffusion equation is solved with a nodal method consistent with that implemented in the licensed core design code SIMULATE5. Arbitrary number of neutron and delayed neutron precursor groups can be used. For the solution of the spatial problem, the coupling coefficients used to relate the node leakages are found by first converting the time dependent diffusion equation to a static diffusion equation with the use of flux and delayed neutron precursor dynamic frequencies. Once the static-like equations are obtained, the multi-group analytical nodal model is used to obtain the coupling coefficients, expressing the node leakage in terms of adjacent node average fluxes. The coupling coefficients are then inserted into the time dependent nodal balance equation. For the time integration, the time dependent neutron balance equation is solved with the frequency transformation method. The treatment of the temporal dependence yields a fixed source problem which can be solved utilizing the existing fixed-source methodology. The primary purpose of this paper is to describe the neutron kinetics methodology implemented in SIMULATE5-K. The accuracy of the method is demonstrated for a series of well-known, neutronic-only benchmark problems. (author)
A systematic solution approach for the neutron transport equation, based on a least-squares finite-element discretization, is presented. This approach includes the theory for the existence and uniqueness of the analytical as well as of the discrete solution, bounds for the discretization error, and guidance for the development of an efficient multigrid solver for the resulting discrete problem. To guarantee the accuracy of the discrete solution for diffusive regimes, a scaling transformation is applied to the transport operator prior to the discretization. The key result is the proof of the V-ellipticity and continuity of the scaled least-squares bilinear form with constants that are independent of the total cross section and the absorption cross section. For a variety of least-squares finite-element discretizations this leads to error bounds that remain valid in diffusive regimes. Moreover, for problems in slab geometry a full multigrid solver is presented with V(1, 1)-cycle convergence rates approximately equal to 0.1, independent of the size of the total cross section and the absorption cross section.